基于matlab的RBF神经网络的非线性方程回归实现
1 引言
1.1研究背景及意义
实际应用结果表明, 该神经网络可以很好地克服BP 神经网络学习过程的收敛过分 依赖于初值和可能出现局部收敛的缺陷, 具有较快的运算速度、较强的非线性映射能力和较好的效能。RBF径向基函数，取值仅依赖于离原点距离的实值函数，任何满足这种特性的函数就叫径向基函数。一般采用欧氏距离，也可以用其他距离。RBF的中心点确定以后，这种映射关系也就确定了。而隐含层空间到输出空间的映射是线性的，即网络的输出是隐单元输出的线性加权和，此处的权即为网络可调参数。由此可见，从总体上看，网络由输入到输出的映射是非线性的，而网络输出对可调参数而言却又是线性的。这样网络的权就可由线性方程组直接解出，从而大大加快学习速度并避免局部极小问题。
1.2本文研究内容与结构
本文中，我展示了使用matlab用可视化的方法观察准确RBF神经网络的拟合效果。1.m为严格(Exact)径向基网络来实现非线性的函数回归。2.m为RBF网络对同一函数拟合
2 实验数据
当我们在1.m中观察RBF神经网络拟合效果的原始数据
- m中产生训练数据
- 实验模型与方法
单隐层的前馈神经网络，使用径向基函数作为隐层神经元的激活函数，输出层则是对隐层神经元输出的线性组合（理论上可以设置多个隐层，但一般只设计一个隐层）。所以从输入空间到隐含层空间的变换是非线性的，而从隐含层空间到输出层空间变换是线性的。
根据Cover定理，低维空间不可分的数据到了高维空间会更有可能变得可分。换句话来说，RBF网络的隐层的功能就是将低维空间的输入通过非线性函数映射到一个高维空间，然后再在这个高维空间进行曲线的拟合，它等价于在一个隐含的高维空间寻找一个能最佳拟合训练数据的表面。
当RBF的中心点确定以后，这种映射关系也就确定了。而隐含层空间到输出空间的映射是线性的，即网络的输出是隐单元输出的线性加权和，此处的权即为网络可调参数。由此可见，从总体上看，网络由输入到输出的映射是非线性的，而网络输出对可调参数而言却又是线性的。这样网络的权就可由线性方程组直接解出，从而大大加快学习速度并避免局部极小问题。
从另一个方面也可以这样理解，多层感知器（包括BP神经网络）的隐节点基函数采用线性函数，通过激活函数来增加非线性因素；而RBF网络的隐节点的基函数采用距离函数（如欧氏距离），并使用径向基函数（如Gaussian函数）作为激活函数（bp网络中输入到隐层，隐层到输出都是非线性的，因为都有激活函数）。径向基函数关于n维空间的一个中心点具有径向对称性，而且神经元的输入离该中心点越远，神经元的激活程度就越低（值越小）。
-
- RBF神经网络的训练
1.卷确定神经元的中心，常用有随机采样，聚类（聚类中心当成径向基函数的h个中心）
2.确定参数和阈值，最小均方误差LMS直接计算得到，或者通过梯度下降来对网络中的参数都进行监督训练优化，代价函数是网络输出和期望输出的均方误差
3.只要有足够多的隐层单元，RBF能以任意精度逼近任意连续函数线性的非线性函数。
-
- 径向基函数网络结构
径向基函数(Radial Basis Function, RBF)神经网络是一种单隐含层的三层前馈神经网络，网络结构如下图所示
3.3 实验训练参数及实验环境
（1）可视化拟合 ---1.m%% 网络建立和训练
% 网络建立 输入为[x1;x2],输出为F。Spread使用默认。
net=newrbe([x1;x2],F)
（2）非线性逼近---2.m
%% 产生训练样本（训练输入，训练输出）
% ld为样本例数
ld=400;
% 产生2*ld的矩阵
x=rand(2,ld);
% 将x转换到[-1.5 1.5]之间
x=(x-0.5)*1.5*2;
% x的第一列为x1，第二列为x2.
x1=x(1,:);
x2=x(2,:);
% 计算网络输出F值
F=20+x1.^2-10*cos(2*pi*x1)+x2.^2-10*cos(2*pi*x2);
实验环境为matlab2021a
3.4 RBF神经网络算法
便于分析，记RBF神经网络的输入为m维，隐含层有s1个神经元，输出层有s2个神经元。RBF神经网络隐含神经元的激活函数为如下所示的高斯基函数
为第j个隐含层神经元高斯基函数的中心向量， m表示网络输入的x维数，bj为第j个隐含层神经元高斯基函数的宽度。与之前提及的BP神经网络中的sigmod函数不同，高斯基函数只在有限的范围内，输入是非零的，超过一定的范围，其输出则为零。
b j越大，高斯基函数的非零输出区域越大，表明对输入的映射能力越强。
Cj表示非零输出区域的中心位置，输入离中心越近，其输出值会更大，表明高斯基函数对输入更加敏感。
- 实验结论
4.1采用RBF神经网络逼近非线性函数
可知，采用的RBF神经网络输入和输出层神经元数量都为1，设置隐含层神经元数量为10个。
由于RBF神经网络采用的激活函数在有限区域内输出为非零，因此在设计RBF神经网络的时候，需要根据网络的输入来确定每个隐含层神经元激活函数的参数，合适的网络参数能够提升网络的逼近效果和性能，而不合适的参数会导致神经网络训练失败，或者达不到想要的效果。
通常来说，参数需要与网络的输入相匹配，要保证输入在高斯基函数的有效映射区域内；同样的，根据输入的范围和高斯基函数的中心，来设置一个合适的宽度参数。
4.2神经网络拟合可视化结果
%% 网络的效果验证
% 我们将原数据回带，测试网络效果：
ty=sim(net,[x1;x2]);
% 我们使用图像来看网络对非线性函数的拟合效果
figure
plot3(x1,x2,F,'rd');
hold on;
plot3(x1,x2,ty,'b-.');
view(113,36)
title('可视化的方法观察准确RBF神经网络的拟合效果')
xlabel('x1')
ylabel('x2')
zlabel('F')
grid on
4.3神经网络逼近结果
4.3.1 训练过程中收敛情况
采用approximate RBF神经网络。spread为默认值
net=newrb(x,F);
NEWRB, neurons = 0, MSE = 108.563
NEWRB, neurons = 2, MSE = 106.612
NEWRB, neurons = 3, MSE = 106.588
NEWRB, neurons = 4, MSE = 105.692
NEWRB, neurons = 5, MSE = 103.957
NEWRB, neurons = 6, MSE = 103.901
NEWRB, neurons = 7, MSE = 103.661
NEWRB, neurons = 8, MSE = 99.4239
NEWRB, neurons = 9, MSE = 99.3709
NEWRB, neurons = 10, MSE = 98.4437
NEWRB, neurons = 11, MSE = 98.4334
NEWRB, neurons = 12, MSE = 97.0947
NEWRB, neurons = 13, MSE = 97.0662
NEWRB, neurons = 14, MSE = 97.0623
NEWRB, neurons = 15, MSE = 96.9707
NEWRB, neurons = 16, MSE = 96.5273
NEWRB, neurons = 17, MSE = 89.0728
NEWRB, neurons = 18, MSE = 83.7398
NEWRB, neurons = 19, MSE = 82.6886
NEWRB, neurons = 20, MSE = 73.5795
NEWRB, neurons = 21, MSE = 68.2986
NEWRB, neurons = 22, MSE = 64.3154
NEWRB, neurons = 23, MSE = 56.6795
NEWRB, neurons = 24, MSE = 56.2971
NEWRB, neurons = 25, MSE = 48.7566
NEWRB, neurons = 26, MSE = 47.2739
NEWRB, neurons = 27, MSE = 45.9051
NEWRB, neurons = 28, MSE = 45.0112
NEWRB, neurons = 29, MSE = 42.8181
NEWRB, neurons = 30, MSE = 41.491
NEWRB, neurons = 31, MSE = 35.6978
NEWRB, neurons = 32, MSE = 35.0401
NEWRB, neurons = 33, MSE = 31.7462
NEWRB, neurons = 34, MSE = 30.1603
NEWRB, neurons = 35, MSE = 29.2863
NEWRB, neurons = 36, MSE = 28.3875
NEWRB, neurons = 37, MSE = 26.2179
NEWRB, neurons = 38, MSE = 24.2333
NEWRB, neurons = 39, MSE = 21.639
NEWRB, neurons = 40, MSE = 20.144
NEWRB, neurons = 41, MSE = 18.4123
NEWRB, neurons = 42, MSE = 15.9317
NEWRB, neurons = 43, MSE = 14.5087
NEWRB, neurons = 44, MSE = 11.1312
NEWRB, neurons = 45, MSE = 8.44281
NEWRB, neurons = 46, MSE = 6.90717
NEWRB, neurons = 47, MSE = 5.66637
NEWRB, neurons = 48, MSE = 4.84694
NEWRB, neurons = 49, MSE = 4.38789
NEWRB, neurons = 50, MSE = 3.67744
NEWRB, neurons = 51, MSE = 3.51838
NEWRB, neurons = 52, MSE = 3.23929
NEWRB, neurons = 53, MSE = 2.91933
NEWRB, neurons = 54, MSE = 2.5525
NEWRB, neurons = 55, MSE = 2.07774
NEWRB, neurons = 56, MSE = 1.58034
NEWRB, neurons = 57, MSE = 1.4921
NEWRB, neurons = 58, MSE = 1.04437
NEWRB, neurons = 59, MSE = 0.968894
NEWRB, neurons = 60, MSE = 0.874836
NEWRB, neurons = 61, MSE = 0.811536
NEWRB, neurons = 62, MSE = 0.774263
NEWRB, neurons = 63, MSE = 0.739684
NEWRB, neurons = 64, MSE = 0.63243
NEWRB, neurons = 65, MSE = 0.595342
NEWRB, neurons = 66, MSE = 0.578419
NEWRB, neurons = 67, MSE = 0.513412
NEWRB, neurons = 68, MSE = 0.431057
NEWRB, neurons = 69, MSE = 0.348383
NEWRB, neurons = 70, MSE = 0.28326
NEWRB, neurons = 71, MSE = 0.260364
NEWRB, neurons = 72, MSE = 0.233631
NEWRB, neurons = 73, MSE = 0.17438
NEWRB, neurons = 74, MSE = 0.140672
NEWRB, neurons = 75, MSE = 0.127232
NEWRB, neurons = 76, MSE = 0.107695
NEWRB, neurons = 77, MSE = 0.0939278
NEWRB, neurons = 78, MSE = 0.0755751
NEWRB, neurons = 79, MSE = 0.0680948
NEWRB, neurons = 80, MSE = 0.0523434
NEWRB, neurons = 81, MSE = 0.0433914
NEWRB, neurons = 82, MSE = 0.0417353
NEWRB, neurons = 83, MSE = 0.0358314
NEWRB, neurons = 84, MSE = 0.0308747
NEWRB, neurons = 85, MSE = 0.0227491
NEWRB, neurons = 86, MSE = 0.0191109
NEWRB, neurons = 87, MSE = 0.016148
NEWRB, neurons = 88, MSE = 0.01186
NEWRB, neurons = 89, MSE = 0.00995707
NEWRB, neurons = 90, MSE = 0.00970474
NEWRB, neurons = 91, MSE = 0.00692245
NEWRB, neurons = 92, MSE = 0.00604538
NEWRB, neurons = 93, MSE = 0.00536777
NEWRB, neurons = 94, MSE = 0.0040162
NEWRB, neurons = 95, MSE = 0.00339511
NEWRB, neurons = 96, MSE = 0.00291372
NEWRB, neurons = 97, MSE = 0.0025272
NEWRB, neurons = 98, MSE = 0.002166
NEWRB, neurons = 99, MSE = 0.0018282
NEWRB, neurons = 100, MSE = 0.00159732
NEWRB, neurons = 101, MSE = 0.00146625
NEWRB, neurons = 102, MSE = 0.00127549
NEWRB, neurons = 103, MSE = 0.00112714
NEWRB, neurons = 104, MSE = 0.00061357
NEWRB, neurons = 105, MSE = 0.000556239
NEWRB, neurons = 106, MSE = 0.000511452
NEWRB, neurons = 107, MSE = 0.00039547
NEWRB, neurons = 108, MSE = 0.000414003
NEWRB, neurons = 109, MSE = 0.000375354
NEWRB, neurons = 110, MSE = 0.000372545
NEWRB, neurons = 111, MSE = 0.000363626
NEWRB, neurons = 112, MSE = 0.000362367
NEWRB, neurons = 113, MSE = 0.0003299
NEWRB, neurons = 114, MSE = 0.00032986
NEWRB, neurons = 115, MSE = 0.000322453
NEWRB, neurons = 116, MSE = 0.000318573
NEWRB, neurons = 117, MSE = 0.000277645
NEWRB, neurons = 118, MSE = 0.000211484
NEWRB, neurons = 119, MSE = 0.000211489
NEWRB, neurons = 120, MSE = 0.000201869
NEWRB, neurons = 121, MSE = 0.0001469
NEWRB, neurons = 122, MSE = 0.000146576
NEWRB, neurons = 123, MSE = 0.000112525
NEWRB, neurons = 124, MSE = 0.000108957
NEWRB, neurons = 125, MSE = 8.90606e-005
NEWRB, neurons = 126, MSE = 8.95122e-005
NEWRB, neurons = 127, MSE = 7.56872e-005
NEWRB, neurons = 128, MSE = 8.23662e-005
NEWRB, neurons = 129, MSE = 8.13812e-005
NEWRB, neurons = 130, MSE = 6.45973e-005
NEWRB, neurons = 131, MSE = 6.44272e-005
NEWRB, neurons = 132, MSE = 7.84435e-005
NEWRB, neurons = 133, MSE = 7.41849e-005
NEWRB, neurons = 134, MSE = 7.50613e-005
NEWRB, neurons = 135, MSE = 6.9601e-005
NEWRB, neurons = 136, MSE = 7.24983e-005
NEWRB, neurons = 137, MSE = 6.32684e-005
NEWRB, neurons = 138, MSE = 5.44679e-005
NEWRB, neurons = 139, MSE = 5.62032e-005
NEWRB, neurons = 140, MSE = 4.48197e-005
NEWRB, neurons = 141, MSE = 4.48237e-005
NEWRB, neurons = 142, MSE = 4.48131e-005
NEWRB, neurons = 143, MSE = 5.18121e-005
NEWRB, neurons = 144, MSE = 5.06992e-005
NEWRB, neurons = 145, MSE = 4.71318e-005
NEWRB, neurons = 146, MSE = 5.10829e-005
NEWRB, neurons = 147, MSE = 4.38683e-005
NEWRB, neurons = 148, MSE = 4.31517e-005
NEWRB, neurons = 149, MSE = 4.31047e-005
NEWRB, neurons = 150, MSE = 2.661e-005
NEWRB, neurons = 151, MSE = 2.49899e-005
NEWRB, neurons = 152, MSE = 2.50251e-005
NEWRB, neurons = 153, MSE = 3.22083e-005
NEWRB, neurons = 154, MSE = 3.12949e-005
NEWRB, neurons = 155, MSE = 3.23248e-005
NEWRB, neurons = 156, MSE = 3.25071e-005
NEWRB, neurons = 157, MSE = 3.29195e-005
NEWRB, neurons = 158, MSE = 3.94727e-005
NEWRB, neurons = 159, MSE = 3.94037e-005
NEWRB, neurons = 160, MSE = 3.84722e-005
NEWRB, neurons = 161, MSE = 1.8859e-005
NEWRB, neurons = 162, MSE = 1.88555e-005
NEWRB, neurons = 163, MSE = 1.53866e-005
NEWRB, neurons = 164, MSE = 1.40733e-005
NEWRB, neurons = 165, MSE = 1.40568e-005
NEWRB, neurons = 166, MSE = 1.65526e-005
NEWRB, neurons = 167, MSE = 9.53138e-006
NEWRB, neurons = 168, MSE = 9.53355e-006
NEWRB, neurons = 169, MSE = 6.22108e-006
NEWRB, neurons = 170, MSE = 6.0591e-006
NEWRB, neurons = 171, MSE = 5.18807e-006
NEWRB, neurons = 172, MSE = 8.00407e-006
NEWRB, neurons = 173, MSE = 6.26717e-006
NEWRB, neurons = 174, MSE = 6.18919e-006
NEWRB, neurons = 175, MSE = 6.19012e-006
NEWRB, neurons = 176, MSE = 6.18806e-006
NEWRB, neurons = 177, MSE = 5.84342e-006
NEWRB, neurons = 178, MSE = 5.82603e-006
NEWRB, neurons = 179, MSE = 5.82554e-006
NEWRB, neurons = 180, MSE = 5.82447e-006
NEWRB, neurons = 181, MSE = 5.82541e-006
NEWRB, neurons = 182, MSE = 4.93829e-006
NEWRB, neurons = 183, MSE = 4.75276e-006
NEWRB, neurons = 184, MSE = 5.26563e-006
NEWRB, neurons = 185, MSE = 5.26646e-006
NEWRB, neurons = 186, MSE = 5.26758e-006
NEWRB, neurons = 187, MSE = 5.26861e-006
NEWRB, neurons = 188, MSE = 5.26744e-006
NEWRB, neurons = 189, MSE = 5.26574e-006
NEWRB, neurons = 190, MSE = 5.26649e-006
NEWRB, neurons = 191, MSE = 5.26647e-006
NEWRB, neurons = 192, MSE = 5.2696e-006
NEWRB, neurons = 193, MSE = 5.2676e-006
NEWRB, neurons = 194, MSE = 5.26874e-006
NEWRB, neurons = 195, MSE = 5.2682e-006
NEWRB, neurons = 196, MSE = 5.26588e-006
NEWRB, neurons = 197, MSE = 5.26638e-006
NEWRB, neurons = 198, MSE = 2.40265e-006
NEWRB, neurons = 199, MSE = 2.39407e-006
NEWRB, neurons = 200, MSE = 2.39367e-006
NEWRB, neurons = 201, MSE = 2.39277e-006
NEWRB, neurons = 202, MSE = 1.87411e-006
NEWRB, neurons = 203, MSE = 1.87375e-006
NEWRB, neurons = 204, MSE = 1.46317e-006
NEWRB, neurons = 205, MSE = 7.67089e-007
NEWRB, neurons = 206, MSE = 1.17599e-006
NEWRB, neurons = 207, MSE = 5.98909e-007
NEWRB, neurons = 208, MSE = 1.80047e-006
NEWRB, neurons = 209, MSE = 1.64662e-006
NEWRB, neurons = 210, MSE = 1.64716e-006
NEWRB, neurons = 211, MSE = 1.77425e-006
NEWRB, neurons = 212, MSE = 1.77418e-006
NEWRB, neurons = 213, MSE = 1.77498e-006
NEWRB, neurons = 214, MSE = 7.71438e-007
NEWRB, neurons = 215, MSE = 5.97916e-007
NEWRB, neurons = 216, MSE = 5.97961e-007
NEWRB, neurons = 217, MSE = 5.98152e-007
NEWRB, neurons = 218, MSE = 1.38733e-006
NEWRB, neurons = 219, MSE = 1.50115e-006
NEWRB, neurons = 220, MSE = 1.08329e-006
NEWRB, neurons = 221, MSE = 1.19279e-006
NEWRB, neurons = 222, MSE = 1.19269e-006
NEWRB, neurons = 223, MSE = 1.19254e-006
NEWRB, neurons = 224, MSE = 1.19367e-006
NEWRB, neurons = 225, MSE = 2.06955e-006
NEWRB, neurons = 226, MSE = 8.78231e-007
NEWRB, neurons = 227, MSE = 1.63759e-006
NEWRB, neurons = 228, MSE = 1.63009e-006
NEWRB, neurons = 229, MSE = 1.63032e-006
NEWRB, neurons = 230, MSE = 1.63e-006
NEWRB, neurons = 231, MSE = 2.29273e-006
NEWRB, neurons = 232, MSE = 2.29247e-006
NEWRB, neurons = 233, MSE = 6.33408e-007
NEWRB, neurons = 234, MSE = 6.33662e-007
NEWRB, neurons = 235, MSE = 6.52965e-007
NEWRB, neurons = 236, MSE = 8.70292e-007
NEWRB, neurons = 237, MSE = 9.88136e-007
NEWRB, neurons = 238, MSE = 9.49195e-007
NEWRB, neurons = 239, MSE = 6.7905e-007
NEWRB, neurons = 240, MSE = 5.52684e-007
NEWRB, neurons = 241, MSE = 5.52773e-007
NEWRB, neurons = 242, MSE = 8.93576e-007
NEWRB, neurons = 243, MSE = 8.93678e-007
NEWRB, neurons = 244, MSE = 1.26933e-006
NEWRB, neurons = 245, MSE = 1.01884e-006
NEWRB, neurons = 246, MSE = 1.66353e-006
NEWRB, neurons = 247, MSE = 1.66325e-006
NEWRB, neurons = 248, MSE = 1.66364e-006
NEWRB, neurons = 249, MSE = 1.13301e-006
NEWRB, neurons = 250, MSE = 1.16462e-006
NEWRB, neurons = 251, MSE = 6.36004e-007
NEWRB, neurons = 252, MSE = 7.44842e-007
NEWRB, neurons = 253, MSE = 7.44852e-007
NEWRB, neurons = 254, MSE = 7.45151e-007
NEWRB, neurons = 255, MSE = 7.45083e-007
NEWRB, neurons = 256, MSE = 7.4493e-007
NEWRB, neurons = 257, MSE = 6.5226e-007
NEWRB, neurons = 258, MSE = 4.04951e-007
NEWRB, neurons = 259, MSE = 3.9181e-007
NEWRB, neurons = 260, MSE = 5.55604e-007
NEWRB, neurons = 261, MSE = 5.16398e-007
NEWRB, neurons = 262, MSE = 4.80496e-007
NEWRB, neurons = 263, MSE = 5.29451e-007
NEWRB, neurons = 264, MSE = 3.1142e-007
NEWRB, neurons = 265, MSE = 3.12048e-007
NEWRB, neurons = 266, MSE = 4.51416e-007
NEWRB, neurons = 267, MSE = 4.81815e-007
NEWRB, neurons = 268, MSE = 4.81784e-007
NEWRB, neurons = 269, MSE = 4.52286e-007
NEWRB, neurons = 270, MSE = 4.10046e-007
NEWRB, neurons = 271, MSE = 4.39783e-007
NEWRB, neurons = 272, MSE = 3.40396e-007
NEWRB, neurons = 273, MSE = 3.40436e-007
NEWRB, neurons = 274, MSE = 3.98623e-007
NEWRB, neurons = 275, MSE = 3.98338e-007
NEWRB, neurons = 276, MSE = 3.9845e-007
NEWRB, neurons = 277, MSE = 3.98546e-007
NEWRB, neurons = 278, MSE = 3.9847e-007
NEWRB, neurons = 279, MSE = 3.98492e-007
NEWRB, neurons = 280, MSE = 3.98618e-007
NEWRB, neurons = 281, MSE = 5.42019e-007
NEWRB, neurons = 282, MSE = 5.61352e-007
NEWRB, neurons = 283, MSE = 5.61104e-007
NEWRB, neurons = 284, MSE = 4.07669e-007
NEWRB, neurons = 285, MSE = 4.97986e-007
NEWRB, neurons = 286, MSE = 4.98001e-007
NEWRB, neurons = 287, MSE = 4.37486e-007
NEWRB, neurons = 288, MSE = 4.37475e-007
NEWRB, neurons = 289, MSE = 2.35071e-007
NEWRB, neurons = 290, MSE = 2.30437e-007
NEWRB, neurons = 291, MSE = 2.34569e-007
NEWRB, neurons = 292, MSE = 2.19626e-007
NEWRB, neurons = 293, MSE = 4.20924e-007
NEWRB, neurons = 294, MSE = 3.26139e-007
NEWRB, neurons = 295, MSE = 2.36303e-007
NEWRB, neurons = 296, MSE = 2.3628e-007
NEWRB, neurons = 297, MSE = 2.43922e-007
NEWRB, neurons = 298, MSE = 2.44038e-007
NEWRB, neurons = 299, MSE = 2.43914e-007
NEWRB, neurons = 300, MSE = 2.44027e-007
NEWRB, neurons = 301, MSE = 2.43918e-007
NEWRB, neurons = 302, MSE = 2.87254e-007
NEWRB, neurons = 303, MSE = 3.51538e-007
NEWRB, neurons = 304, MSE = 3.51618e-007
NEWRB, neurons = 305, MSE = 3.51608e-007
NEWRB, neurons = 306, MSE = 3.5166e-007
NEWRB, neurons = 307, MSE = 3.51626e-007
NEWRB, neurons = 308, MSE = 3.51622e-007
NEWRB, neurons = 309, MSE = 3.63887e-007
NEWRB, neurons = 310, MSE = 3.63839e-007
NEWRB, neurons = 311, MSE = 5.2455e-007
NEWRB, neurons = 312, MSE = 2.66422e-007
NEWRB, neurons = 313, MSE = 1.20945e-007
NEWRB, neurons = 314, MSE = 1.04519e-007
NEWRB, neurons = 315, MSE = 9.70747e-008
NEWRB, neurons = 316, MSE = 9.70575e-008
NEWRB, neurons = 317, MSE = 9.70639e-008
NEWRB, neurons = 318, MSE = 1.07767e-007
NEWRB, neurons = 319, MSE = 9.24604e-008
NEWRB, neurons = 320, MSE = 1.02386e-007
NEWRB, neurons = 321, MSE = 1.024e-007
NEWRB, neurons = 322, MSE = 9.63013e-008
NEWRB, neurons = 323, MSE = 9.62991e-008
NEWRB, neurons = 324, MSE = 9.62671e-008
NEWRB, neurons = 325, MSE = 7.61319e-008
NEWRB, neurons = 326, MSE = 7.61482e-008
NEWRB, neurons = 327, MSE = 7.61234e-008
NEWRB, neurons = 328, MSE = 7.61384e-008
NEWRB, neurons = 329, MSE = 7.29223e-008
NEWRB, neurons = 330, MSE = 7.54182e-008
NEWRB, neurons = 331, MSE = 7.53905e-008
NEWRB, neurons = 332, MSE = 5.26316e-008
NEWRB, neurons = 333, MSE = 1.0217e-007
NEWRB, neurons = 334, MSE = 1.02161e-007
NEWRB, neurons = 335, MSE = 5.85843e-008
NEWRB, neurons = 336, MSE = 5.85873e-008
NEWRB, neurons = 337, MSE = 5.85871e-008
NEWRB, neurons = 338, MSE = 5.98664e-008
NEWRB, neurons = 339, MSE = 5.4952e-008
NEWRB, neurons = 340, MSE = 5.88298e-008
NEWRB, neurons = 341, MSE = 5.17393e-008
NEWRB, neurons = 342, MSE = 6.14429e-008
NEWRB, neurons = 343, MSE = 7.93672e-008
NEWRB, neurons = 344, MSE = 6.31095e-008
NEWRB, neurons = 345, MSE = 7.01942e-008
NEWRB, neurons = 346, MSE = 7.71407e-008
NEWRB, neurons = 347, MSE = 7.45136e-008
NEWRB, neurons = 348, MSE = 6.3065e-008
NEWRB, neurons = 349, MSE = 6.52697e-008
NEWRB, neurons = 350, MSE = 6.31525e-008
NEWRB, neurons = 351, MSE = 7.44494e-008
NEWRB, neurons = 352, MSE = 6.81998e-008
NEWRB, neurons = 353, MSE = 5.07287e-008
NEWRB, neurons = 354, MSE = 5.39876e-008
NEWRB, neurons = 355, MSE = 5.11419e-008
NEWRB, neurons = 356, MSE = 5.35645e-008
NEWRB, neurons = 357, MSE = 6.11236e-008
NEWRB, neurons = 358, MSE = 5.14069e-008
NEWRB, neurons = 359, MSE = 5.35963e-008
NEWRB, neurons = 360, MSE = 5.35946e-008
NEWRB, neurons = 361, MSE = 5.35826e-008
NEWRB, neurons = 362, MSE = 5.35929e-008
NEWRB, neurons = 363, MSE = 4.99915e-008
NEWRB, neurons = 364, MSE = 6.16603e-008
NEWRB, neurons = 365, MSE = 6.31775e-008
NEWRB, neurons = 366, MSE = 6.31942e-008
NEWRB, neurons = 367, MSE = 6.58402e-008
NEWRB, neurons = 368, MSE = 5.4157e-008
NEWRB, neurons = 369, MSE = 5.41499e-008
NEWRB, neurons = 370, MSE = 5.41546e-008
NEWRB, neurons = 371, MSE = 5.91478e-008
NEWRB, neurons = 372, MSE = 5.91258e-008
NEWRB, neurons = 373, MSE = 5.91379e-008
NEWRB, neurons = 374, MSE = 6.87742e-008
NEWRB, neurons = 375, MSE = 6.56028e-008
NEWRB, neurons = 376, MSE = 6.56144e-008
NEWRB, neurons = 377, MSE = 6.56108e-008
NEWRB, neurons = 378, MSE = 6.56184e-008
NEWRB, neurons = 379, MSE = 6.56057e-008
NEWRB, neurons = 380, MSE = 8.2516e-008
NEWRB, neurons = 381, MSE = 8.25455e-008
NEWRB, neurons = 382, MSE = 8.25012e-008
NEWRB, neurons = 383, MSE = 8.25189e-008
NEWRB, neurons = 384, MSE = 4.51011e-008
NEWRB, neurons = 385, MSE = 5.29263e-008
NEWRB, neurons = 386, MSE = 5.29392e-008
NEWRB, neurons = 387, MSE = 5.08806e-008
NEWRB, neurons = 388, MSE = 5.0887e-008
NEWRB, neurons = 389, MSE = 5.08651e-008
NEWRB, neurons = 390, MSE = 5.08715e-008
NEWRB, neurons = 391, MSE = 4.6072e-008
NEWRB, neurons = 392, MSE = 4.60703e-008
NEWRB, neurons = 393, MSE = 4.60743e-008
NEWRB, neurons = 394, MSE = 4.60913e-008
NEWRB, neurons = 395, MSE = 6.03445e-008
NEWRB, neurons = 396, MSE = 5.2683e-008
NEWRB, neurons = 397, MSE = 6.28863e-008
NEWRB, neurons = 398, MSE = 5.9419e-008
NEWRB, neurons = 399, MSE = 2.05435e-007
NEWRB, neurons = 400, MSE = 6.41196e-008
4.3.2 使用RBF网络进行模拟
使用建立的RBF网络进行模拟，得出网络输出
%% 使用图像，画出3维图
% 真正的函数图像
interval=0.1;
[x1, x2]=meshgrid(-1.5:interval:1.5);
F = 20+x1.^2-10*cos(2*pi*x1)+x2.^2-10*cos(2*pi*x2);
subplot(1,3,1)
mesh(x1,x2,F);
zlim([0,60])
title('真正的函数图像')
% 网络得出的函数图像
v=reshape(ty,row);
subplot(1,3,2)
mesh(i,j,v);
zlim([0,60])
title('RBF神经网络结果')
% 误差图像
subplot(1,3,3)
mesh(x1,x2,F-v);
zlim([0,60])
title('误差图像')
set(gcf,'position',[300 ,250,900,400])
5.实验代码
5.1 1.m
%% 清空环境变量
clc
clear
%% 产生输入 输出数据
% 设置步长
interval=0.01;
% 产生x1 x2
x1=-1.5:interval:1.5;
x2=-1.5:interval:1.5;
% 按照函数先求得相应的函数值，作为网络的输出。
F =20+x1.^2-10*cos(2*pi*x1)+x2.^2-10*cos(2*pi*x2);
%% 网络建立和训练
% 网络建立 输入为[x1;x2],输出为F。Spread使用默认。
net=newrbe([x1;x2],F)
%% 网络的效果验证
% 我们将原数据回带，测试网络效果：
ty=sim(net,[x1;x2]);
% 我们使用图像来看网络对非线性函数的拟合效果
figure
plot3(x1,x2,F,'rd');
hold on;
plot3(x1,x2,ty,'b-.');
view(113,36)
title('可视化的方法观察准确RBF神经网络的拟合效果')
xlabel('x1')
ylabel('x2')
zlabel('F')
grid on
web browser https://blog.csdn.net/NREsa
5.2 2.m
%% 清空环境变量
clc
clear
%% 产生训练样本（训练输入，训练输出）
% ld为样本例数
ld=400;
% 产生2*ld的矩阵
x=rand(2,ld);
% 将x转换到[-1.5 1.5]之间
x=(x-0.5)*1.5*2;
% x的第一列为x1，第二列为x2.
x1=x(1,:);
x2=x(2,:);
% 计算网络输出F值
F=20+x1.^2-10*cos(2*pi*x1)+x2.^2-10*cos(2*pi*x2);
%% 建立RBF神经网络
% 采用approximate RBF神经网络。spread为默认值
net=newrb(x,F);
%% 建立测试样本
% generate the testing data
interval=0.1;
[i, j]=meshgrid(-1.5:interval:1.5);
row=size(i);
tx1=i(:);
tx1=tx1';
tx2=j(:);
tx2=tx2';
tx=[tx1;tx2];
%% 使用建立的RBF网络进行模拟，得出网络输出
ty=sim(net,tx);
%% 使用图像，画出3维图
% 真正的函数图像
interval=0.1;
[x1, x2]=meshgrid(-1.5:interval:1.5);
F = 20+x1.^2-10*cos(2*pi*x1)+x2.^2-10*cos(2*pi*x2);
subplot(1,3,1)
mesh(x1,x2,F);
zlim([0,60])
title('真正的函数图像')
% 网络得出的函数图像
v=reshape(ty,row);
subplot(1,3,2)
mesh(i,j,v);
zlim([0,60])
title('RBF神经网络结果')
% 误差图像
subplot(1,3,3)
mesh(x1,x2,F-v);
zlim([0,60])
title('误差图像')
set(gcf,'position',[300 ,250,900,400])
6 结语
RBF神将网络是一种三层神经网络，其包括输入层、隐层、输出层。从输入空间到隐层空间的变换是非线性的，而从隐层空间到输出层空间变换是线性的. RBF网络的基本思想是：用RBF作为隐单元的“基”构成隐含层空间，这样就可以将输入矢量直接映射到隐空间，而不需要通过权连接。当RBF的中心点确定以后，这种映射关系也就确定了。而隐含层空间到输出空间的映射是线性的，即网络的输出是隐单元输出的线性加权和，此处的权即为网络可调参数。其中，隐含层的作用是把向量从低维度的p映射到高维度的h，这样低维度线性不可分的情况到高维度就可以变得线性可分了，主要就是核函数的思想。这样，网络由输入到输出的映射是非线性的，而网络输出对可调参数而言却又是线性的。网络的权就可由线性方程组直接解出，从而大大加快学习速度并避免局部极小问题。